A ball in the product metric for finite index set is the Cartesian
product of balls in all coordinates. For infinite index set this is no
longer true; instead the correct statement is that a *closed ball* is
the product of closed balls in each coordinate (where closed ball means
a set of the form in blcld18540) - for a counterexample the point in
whose -th coordinate is is in
but is not in
the -ball of the
product (since ).

The last assumption, , is needed only in the
case
, when the right side evaluates to
and the left
evaluates to if and if
.
(Contributed by Mario Carneiro, 28-Aug-2015.)

The neighborhoods around a point of a metric space are those
subsets containing a ball around . Definition of neighborhood in
[Kreyszig] p. 19. (Contributed by NM,
8-Nov-2007.) (Revised by Mario
Carneiro, 23-Dec-2013.)

The closure of an open ball in a metric space is contained in the
corresponding closed ball. (The converse is not, in general, true; for
example, with the discrete metric, the closed ball of radius 1 is the
whole space, but the open ball of radius 1 is just a point, whose
closure is also a point.) (Contributed by Mario Carneiro,
31-Dec-2013.)

Two ways of saying that two metrics generate the same topology. Two
metrics satisfying the right-hand side are said to be (topologically)
equivalent. (Contributed by Jeffrey Hankins, 21-Jun-2009.) (Revised by
Mario Carneiro, 12-Nov-2013.)

If the metric is
"strongly finer" than (meaning that there
is a positive real constant such that
), then generates a finer
topology. (Using this theorem twice in each direction states that if
two metrics are strongly equivalent, then they generate the same
topology.) (Contributed by Mario Carneiro, 14-Sep-2015.)

The topology generated by an extended metric can also be generated by a
true metric. Thus, "metrizable topologies" can equivalently
be defined
in terms of metrics or extended metrics. (Contributed by Mario
Carneiro, 26-Aug-2015.)

The indexed product structure is an extended metric space when the index
set is finite. (Although the extended metric is still valid when the
index set is infinite, it no longer agrees with the product topology,
which is not metrizable in any case.) (Contributed by Mario Carneiro,
28-Aug-2015.)

Two ways to say a mapping from metric to metric is
continuous at point . The distance arguments are swapped compared
to metcnp18576 (and Munkres' metcn18578) for compatibility with df-lm17298.
Definition 1.3-3 of [Kreyszig] p. 20.
(Contributed by NM, 4-Jun-2007.)
(Revised by Mario Carneiro, 13-Nov-2013.)

Two ways to say a mapping from metric to metric is
continuous. Theorem 10.1 of [Munkres]
p. 127. The second biconditional
argument says that for every positive "epsilon" there is a
positive "delta" such that a distance less than delta in
maps to a distance less than epsilon in . (Contributed by NM,
15-May-2007.) (Revised by Mario Carneiro, 28-Aug-2015.)

Any two elements of the filter base generated by the metric can be
compared, like for RR+ (i.e. it's totally ordered). (Contributed by
Thierry Arnoux, 22-Nov-2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

Any two elements of the filter base generated by the metric can be
compared, like for RR+ (i.e. it's totally ordered). (Contributed by
Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux,
11-Feb-2018.)

The identity diagonal is included in all elements of the filter base
generated by the metric . (Contributed by Thierry Arnoux,
22-Nov-2017.) (Proof modification is discouraged.)
(New usage is discouraged.)

For any element of
the filter base generated by the metric ,
the half element (corresponding to half the distance) is also in this
base. (Contributed by Thierry Arnoux, 28-Nov-2017.)
(Proof modification is discouraged.) (New usage is discouraged.)

For any element of
the filter base generated by the metric ,
the half element (corresponding to half the distance) is also in this
base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by
Thierry Arnoux, 11-Feb-2018.)